Question

assume that the weight loss for the first month of a diet program varies between 6 pounds and 12 pounds, and is spread evenly over the range of possibilities, so that there is a uniform distribution. find the probability that the given range of pounds lost is between 8 pounds and 11 pounds.
a. $\frac{1}{4}$
b. $\frac{1}{3}$
c. $\frac{2}{3}$
d. $\frac{1}{2}$

Explanation:

Step1: Determine the formula for uniform distribution probability

For a uniform distribution \(X\sim U(a,b)\), the probability density function is \(f(x)=\frac{1}{b - a}\), and the probability \(P(c<X<d)=\frac{d - c}{b - a}\), where \(a = 6\), \(b=12\), \(c = 8\), \(d = 11\).

Step2: Calculate the probability

Substitute the values into the formula: \(P(8<X<11)=\frac{11 - 8}{12 - 6}\).
First, calculate the numerator: \(11-8 = 3\).
Then, calculate the denominator: \(12 - 6=6\).
So, \(P(8<X<11)=\frac{3}{6}=\frac{1}{2}\).

Answer:

D. \(\frac{1}{2}\)